is the right endpoint of the spectrum \(\sigma \left(\Phi _{A^*A}(H)\right)\) .
Since for every operator \(X\)  acting on \(\mathcal {H}\)  we have \(X\in \mathcal {N}\) 
if and only if \(X^*X\in \mathcal {N}\)  if and only if the right endpoint of \(\sigma (X^*X)\)  is its
eigenvalue (see [1]} and [2]}),
we just need to figure out when \(\lambda _{\max }\)  is (or is not) an eigenvalue of \(A^*A\) .
